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Transmutation techniques and observability for time-discrete approximation schemes of conservative systems

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Abstract

In this article, we consider abstract linear conservative systems and their time-discrete counterparts. Our main result is a representation formula expressing solutions of the continuous model through the solution of the corresponding time-discrete one. As an application, we show how observability properties for the time continuous model yield uniform (with respect to the time-step) observability results for its time-discrete approximation counterparts, provided the initial data are suitably filtered. The main output of this approach is the estimate on the time under which we can guarantee uniform observability for the time-discrete models. Besides, using a reverse representation formula, we also prove that this estimate on the time of uniform observability for the time-discrete models is sharp. We then conclude with some general comments and open problems.

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Author information

Authors and Affiliations

  1. Institut de Mathématiques de Toulouse, UMR5219-Université de Toulouse, CNRS, UPS IMT, 31062, Toulouse Cedex 9, France

    Sylvain Ervedoza

  2. BCAM-Basque Center for Applied Mathematics, Alameda de Mazarredo, 14, 48009, Bilbao, Basque Country, Spain

    Enrique Zuazua

  3. Ikerbasque-Basque Foundation for Science, 48013, Bilbao, Basque Country, Spain

    Enrique Zuazua

Authors
  1. Sylvain Ervedoza
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  2. Enrique Zuazua
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Correspondence to Sylvain Ervedoza.

Additional information

The first author was partially supported by the Agence Nationale de la Recherche (ANR, France), Project CISIFS NT09-437023, and Grant MTM2011-29306 of the MINECO, Spain. Part of this work has been done while he was visiting the BCAM—Basque Center for Applied Mathematics as a Visiting Fellow. The second author was supported by the ERC Advanced Grant FP7-246775 NUMERIWAVES, the Grants PI2010-04 and BERC 2014-2017 of the Basque Government, the Grant FA9550-14-1-0214 of the EOARD-AFOSR, Grants MTM2011- 29306 and SEV-2013-0323 of the MINECO (Spain) and by the ANR-11-LABX- 0040-CIMI within the program ANR-11-IDEX-0002-02 of the CIMI (Centre International de Mathématiques et d’Informatique de Toulouse) Excellence program. This work was finalized while Enrique Zuazua was visiting CIMI.

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Ervedoza, S., Zuazua, E. Transmutation techniques and observability for time-discrete approximation schemes of conservative systems. Numer. Math. 130, 425–466 (2015). https://doi.org/10.1007/s00211-014-0668-3

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  • DOI: https://doi.org/10.1007/s00211-014-0668-3

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